Included angle at B from bearings: If the whole-circle bearing of AB is 190° and that of CB is 260° 30', what is the included angle ABC at station B (the smaller interior angle between BA and BC)?

Introduction / Context:
Converting whole-circle bearings (WCB) to included angles at a traverse station is a routine computation. Careful handling of back bearings is needed when bearings are given for directions that do not originate at the station of interest.

Given Data / Assumptions:

• Bearing of AB (from A to B) = 190°.
• Bearing of CB (from C to B) = 260° 30'.
• Required: included angle at B between BA and BC (smaller angle).

Concept / Approach:

At station B we need bearings of BA and BC. These are the back bearings of AB and CB respectively. Back bearing = fore bearing ± 180° (adjust to 0–360°). Then take the smaller difference between the two resulting bearings to get the interior angle at B.

Step-by-Step Solution:

Verification / Alternative check:

A sketch with rays at 10° and 80°30' from a common origin (B) visibly confirms an acute included angle of 70°30'.

Why Other Options Are Wrong:

99°30' and 80°30' arise from misinterpreting fore/back bearings; 120°30' is unrelated; “None” is incorrect because the computation is definite.

Common Pitfalls:

Forgetting to convert to back bearings at the correct station; taking the reflex angle instead of the smaller interior angle.

Final Answer:

70° 30'